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2019å¹´(å¹³æ31å¹´)度å¼ççå ¬ç«é«æ ¡å ¥è©¦çµããã¾ãããã<br />
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<p>⇒ã<a href="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama.pdf" rel="noopener" target="_blank">2019å¼ççå ¬ç«å ¥è©¦æ°å¦åé¡</a></p>
<h3>ï¼ï¼æ¨æ¬èª¿æ»ãä½ç©ãä½å³ã証æåé¡</h3>
<p>ã\(\large{2}\)</p>
<h4>æ¨æ¬èª¿æ»ã¯æ¯ä¾å¼ã«ããæ¨æ¸¬</h4>
<p>(1)ã\(\color{red}{\fbox{ãå¦æ ¡é¸æåé¡ã®1ã®(7)ã}}\)<br />
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<p>ç½ããããããã«ã®ãã£ããã¨ãªã¬ã³ã¸è²ã®ãã£ããã¨ããã¾ããã<br />
ç½ã¨é»ã®ç¢ç³ã§ãåãã§ãã</p>
<p>ãè¢ã«å ¥ã£ãç½ããã£ã¦<span class="red b">\(\,50\,\)åãªã¬ã³ã¸ãå ãã</span>ã<br />
ã\(\,\color{blue}{30}\,\)ååãåºããã\(\,\color{blue}{6}\,\)åãªã¬ã³ã¸ã ã£ãã<br />
ï¼ãããå ¨ä½ã¨ãªã¬ã³ã¸ã®æ¯çã§ããï¼</p>
<p>ä¸é¨ãåãåºãæ¨æ¬èª¿æ»ã«ãªãã¾ãã®ã§ãæ¨æ¸¬ã®çãã«ãªãã¾ãã</p>
<p>ãã ãããã§ã¯ãªã¬ã³ã¸ã\(\,\color{blue}{50}\,\)åå ããã®ã§ã<br />
ãå ã®è¢ã«å ¥ã£ã¦ããç½ã\(\,\color{red}{x}\,\)åã¨ããã¨å ¨ä½ã¯\(\,\color{red}{x}+\color{blue}{50}\,\)<br />
ã¨ãªã£ã¦ãããã¨ã«æ³¨æãå¿ è¦ã§ãã</p>
<p>ã\(\begin{eqnarray}<br />
30ï¼6&#038;=&#038;(x+50):50\\<br />
6(x+50)&#038;=&#038;50\times 30\\<br />
x+50&#038;=&#038;\frac{50\times 30}{6}\\<br />
x+50&#038;=&#038;250\\<br />
x&#038;=&#038;200<br />
\end{eqnarray}\)</p>
<p>ãã£ã¦ããããã\(\,\underline{ã200ã}ã\,\)å</p>
<p>ããã<br />
ã\(\begin{eqnarray}<br />
30ï¼6&#038;=&#038;x:50\\<br />
6x&#038;=&#038;50\times 30\\<br />
x&#038;=&#038;\frac{50\times 30}{6}\\<br />
&#038;=&#038;250<br />
\end{eqnarray}\)</p>
<p>ã¨ããå ´åã¯ãå ãããªã¬ã³ã¸\(\,50\,\)åãå¼ããªãã¦ã¯ãªãã¾ããã</p>
<h4>ä¸è§éã®ä½ç©ã®æ±ãæ¹</h4>
<p>(2)<br />
ãå³ã®å³ã®ãããªãã¨ããã®ã§ããæ¡ä»¶ããã¹ã¦æ¸ãè¾¼ã¾ãã¦ããããã§ã¯ããã¾ããã<br />
足ããªãé¨åã¯èªåã§æ¸ãè¾¼ãã§ãå³ã®ä¸ã§ã§ããã ãåããæãããã¦ããã¨ããã®ãå³å½¢åé¡ã®åºæ¬ã§ãã</p>
<p>ã\(\color{red}{\fbox{æ¡ä»¶}}\)</p>
<p>ã\(\,\mathrm{AB=BC=BD=6}\,\)<br />
ã\(\,\mathrm{∠ABC=∠ABD=∠CBD=90^{\circ}}\,\)<br />
ã\(\,\mathrm{AP:PD=1:2}\,\)<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama016.jpg" alt="" width="264" height="287" class="aligncenter size-full wp-image-13198" /><br />
æ±ããã®ã¯ä¸è§é\(\,\mathrm{PBCD}\,\)ã®ä½ç©ã§ãã</p>
<p>ä¸è§éã®ä½ç©ãæ±ããã«ã¯åºé¢ãé«ããåãã£ã¦ãã¾ããã<br />
ãªã®ã§å ¬å¼ã¯ç´æ¥ä½¿ããããããã¾ããã</p>
<p>å ¬å¼ã使ããªãå ´åã®é¢ç©ãä½ç©ãæ±ããã¨ãã¯ã<br />
ããé¨åãï¼ãé¨åã<br />
ããå ¨ä½ãï¼ãé¨åã<br />
ãèããã¨ãã¾ããããã¨ãå¤ãã§ãã</p>
<p>å ãã¯å ¨ä½ï¼ä¸è§é\(\,\mathrm{ABCD}\,\)ï¼ä½ç©ãåºãã¦ããã¾ãããã</p>
<p>åºé¢ã¯\(\,\mathrm{△ABC}\,\)ã§ã\(\,\mathrm{△BCD}\,\)ã§ãã©ã¡ããç´è§ä¸è§å½¢ã§ã<br />
é«ãã«ãªãåç·ã®é·ããåããã¾ãã®ã§ã©ã¡ãã§ãè¯ãã§ãã</p>
<p>å³ã§ä¸ã«ããã®ã§åºé¢ã\(\,\mathrm{△BCD}\,\)ã¨ãã¾ãããã<br />
ãã®ã¨ãé«ãã¯\(\,\mathrm{AB}\,\)ã§ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama017.jpg" alt="" width="256" height="291" class="aligncenter size-full wp-image-13199" />åºé¢ç©ãåºãã¾ãã</p>
<p>ç«ä½åé¡ãç°¡åã«ããä¸çªã®æ¹æ³ã¯é¢ãæãåºããã¨ã§ãã</p>
<p>\(\,\mathrm{△BCD}\,\)ã¯ç´è§äºç辺ä¸è§å½¢ã§ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama018.jpg" alt="" width="188" height="202" class="aligncenter size-full wp-image-13200" />åºé¢ç©ã¯</p>
<p>ã\(\begin{eqnarray}\displaystyle<br />
\mathrm{△BCD}&#038;=&#038;\frac{1}{2}\times \mathrm{BC}\times \mathrm{BD}\\<br />
&#038;=&#038;\frac{1}{2}\times 6\times 6\\<br />
&#038;=&#038;18<br />
\end{eqnarray}\)</p>
<p>é«ãã¯\(\,\mathrm{AB=6}\,\)ãªã®ã§ä¸è§é\(\,\mathrm{ABCD}\,\)ã®ä½ç©ã¯</p>
<p>ã\(\begin{eqnarray}<br />
\mathrm{ABCD}&#038;=&#038;\frac{1}{3}\times \mathrm{△BCD}\times \mathrm{AB}\\<br />
&#038;=&#038;\frac{1}{3}\times 18\times 6\\<br />
&#038;=&#038;\color{red}{36}<br />
\end{eqnarray}\)</p>
<p>ããããã¯å¤§ãã3ã¤ã®æ¹æ³ã«åããã¾ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama016.jpg" alt="" width="264" height="287" class="aligncenter size-full wp-image-13198" /><br />
ã<span class="red b">ä¸è§é\(\,\mathrm{ABCD}\,\)ããä¸è§é\(\,\mathrm{ABCP}\,\)ãå¼ãã</span><br />
ã<span class="blue b">ä¸è§é\(\,\mathrm{PBCD}\,\)ã®ä½ç©ãç´æ¥æ±ãã«è¡ãã</span><br />
ãããä¸ã¤ã¯ãä¸è§é\(\,\mathrm{ABCD}\,\)ãå§ç¸®ããã</p>
<p>å ¨é¨ãã£ã¦ããã¾ãã</p>
<p>ã<span class="red b">ä¸è§é\(\,\mathrm{ABCD}\,\)ããä¸è§é\(\,\mathrm{ABCP}\,\)ãå¼ãã</span><br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama020.jpg" alt="" width="289" height="323" class="aligncenter size-full wp-image-13202" />ãã®å ´åã¯åºé¢ã\(\,\mathrm{△ABC}\,\)ã§é«ããå³ã®\(\,\mathrm{PH}\,\)ã«ãªãã¾ãã</p>
<p>\(\,\mathrm{PH}\,\)ãå¿ è¦ãªã®ã§\(\,\mathrm{△ABD}\,\)ãæãåºãã¾ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama022.jpg" alt="" width="236" height="210" class="aligncenter size-full wp-image-13204" />\(\,\mathrm{△AHP}\,\)ã∽ã\(\,\mathrm{△ABD}\,\)<br />
ã§<br />
\(\,\mathrm{AP:PD=1:2}\,\)ãªã®ã§<span class="red b">ç¸ä¼¼æ¯ã¯\(\,\mathrm{AH:AB}\,\)ã®\(\,1ï¼3\,\)</span></p>
<p>ãã®ãã¨ãã<br />
ã\(\begin{eqnarray}<br />
1:3&#038;=&#038;\mathrm{PH:DB}\\<br />
3\mathrm{PH}&#038;=&#038;\mathrm{DB}\\<br />
3\mathrm{PH}&#038;=&#038;6\\<br />
\mathrm{PH}&#038;=&#038;2<br />
\end{eqnarray}\)</p>
<p>ãã£ã¦ä¸è§é\(\,\mathrm{PABC}\,\)ã®ä½ç©ã¯</p>
<p>ã\(\begin{eqnarray}\displaystyle<br />
\mathrm{PABC}&#038;=&#038;\frac{1}{3}\times \mathrm{△ABC}\times \mathrm{PH}\\<br />
&#038;=&#038;\frac{1}{3}\times \frac{1}{2}\times 6\times 6\times 2\\<br />
&#038;=&#038;\color{blue}{12}<br />
\end{eqnarray}\)</p>
<p>æ±ããä¸è§é\(\,\mathrm{PBCD}\,\)ã®ä½ç©ã¯</p>
<p>ã\(\begin{eqnarray}\displaystyle<br />
\mathrm{PBCD}&#038;=&#038;\mathrm{ABCD}-\mathrm{PABC}\\<br />
&#038;=&#038;\color{red}{36}-\color{blue}{12}\\<br />
&#038;=&#038;\underline{ã24ã}<br />
\end{eqnarray}\)<br />
ã<br />
ã<span class="blue b">ä¸è§é\(\,\mathrm{PBCD}\,\)ã®ä½ç©ãç´æ¥æ±ãã«è¡ãã</span></p>
<p><img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama023.jpg" alt="" width="242" height="262" class="aligncenter size-full wp-image-13205" />ãã®å ´åã¯åºé¢ã\(\,\mathrm{△BCD}\,\)ãé«ããå³ã®\(\,\mathrm{PT}\,\)ã¨ãã¾ãã</p>
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<p>é«ãã¨ãªã\(\,\mathrm{PT}\,\)ã®é·ãã欲ããã®ã§\(\,\mathrm{△ABD}\,\)ãã¬ãã ãã¾ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama024.jpg" alt="" width="173" height="191" class="aligncenter size-full wp-image-13206" />\(\,\mathrm{△DTP}\,\)ã∽ã\(\,\mathrm{△DBA}\,\)<br />
ã§\(\,\mathrm{AP:PD=1ï¼2}\,\)ãªã®ã§ç¸ä¼¼æ¯ã¨ãªã\(\,\mathrm{DT:DB=PT:AB=2:3}\,\)<br />
ãªã®ã§<br />
ã\(\begin{eqnarray}\displaystyle<br />
2:3&#038;=&#038;\mathrm{PT:AB}\\<br />
3\mathrm{PT}&#038;=&#038;2\mathrm{AB}\\<br />
&#038;=&#038;2\times 6\\<br />
\mathrm{PT}&#038;=&#038;4<br />
\end{eqnarray}\)</p>
<p>ãã£ã¦ä¸è§é\(\,\mathrm{PBCD}\,\)ã®ä½ç©ã¯</p>
<p>ã\(\begin{eqnarray}\displaystyle<br />
\mathrm{PBCD}&#038;=&#038;\frac{1}{3}\times \mathrm{\color{blue}{△BCD}}\times \mathrm{\color{red}{PT}}\\<br />
&#038;=&#038;\frac{1}{3}\times \color{blue}{\frac{1}{2}\times 6\times 6}\times \color{red}{4}\\<br />
&#038;=&#038;\underline{ã24ã}<br />
\end{eqnarray}\)</p>
<p>ããä¸ã¤ã®ä¸è§é\(\,\mathrm{ABCD}\,\)ãå§ç¸®ãããã¨ããã®ã¯ä¸ã®2ã¤ãç°¡åã«ãããã®ã§ãã</p>
<p>ãã£ã¦ãããã¨ã¯åããªã®ã§ä¼å¡åãã«ç°¡åã«è§£çãã¦ããã¾ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama025.jpg" alt="" width="225" height="260" class="aligncenter size-full wp-image-13207" /><br />
\(\,\mathrm{ABCD}\,\)ã®ä½ç©ã¯\(\,36\,\)ã§é ç¹ã\(\,\mathrm{D}\,\)ã¨ã¿ãã¨ãã<br />
ã\(\displaystyle \mathrm{DP=\frac{2}{3}\times DA}\,\)<br />
ã\(\displaystyle \mathrm{DC}=\frac{1}{1}\times \mathrm{DC}\)<br />
ã\(\displaystyle \mathrm{DB}=\frac{1}{1}\times \mathrm{DB}\)</p>
<p>ãªã®ã§æ±ããä¸è§éã®ä½ç©ã¯</p>
<p>ã\(\begin{eqnarray}<br />
\mathrm{PBCD}&#038;=&#038;36\times \frac{2}{3}\times \frac{1}{1}\times \frac{1}{1}\\<br />
&#038;=&#038;\underline{ã24ã}<br />
\end{eqnarray}\)</p>
<p>ãã®åé¡ã¯ãã®ææ³ã¯ä½¿ãã¾ã§ããªãã§ããã<br />
ä¸ã®2ã¤ã¨ãã£ã¦ãããã¨ã¯åãã§ãã</p>
<h4>ä½å³ã®èãæ¹</h4>
<p>(3)ã\(\color{red}{\fbox{ãå¦æ ¡é¸æåé¡ã®2ã®(1)ã}}\)</p>
<p>\(\,105°\,\)ã®ä½å³ã§ãã</p>
<p>ä½å³ã§ä½¿ãã®ã¯åºæ¬çã«2ã¤ã<br />
ãåç·ãå¼ãã<br />
ãè§ã®äºçåç·ãæãã<br />
ã§ãã</p>
<p>ãã ãããã§ã¯\(\,60°\,\)ã®ä½å³ãå¿ è¦ã¨ããã®ã§æ£ä¸è§å½¢ãå©ç¨ãã¾ãã</p>
<p>\(\,105°\,\)ã®ä½å³ã¯ç´æ¥ã§ããªãã®ã§\(\,105°\,\)ã<span class="red b">å解</span>ãããã<br />
åç·ã®\(90°\)ã«\(30°\)ã®äºçåç·ãå ãããã<br />
\(105°\)ã®\(\,2\,\)åã®\(\,210°\,\)ãä½å³ãã¦ããã®è§ã®äºçåç·ãå¼ããã§ãã</p>
<p>ããã§ã¯å解ãå©ç¨ãã¾ããã2ã¤ã®é çªãããã¾ãã</p>
<p>ã\(\,105°=\color{red}{45°}+\color{blue}{60°}\,\)</p>
<p>ã®é çªã§ä½å³ããã</p>
<p>ã\(\,105°=\color{blue}{60°}ï¼\color{red}{45°}\,\)</p>
<p>ã®é çªã§ä½å³ãããã§ãã</p>
<p>ã¤ã¾ãåç·ã®äºçåç·ãå ããæ£ä¸è§å½¢ãå ãã¨ãããã¨ã§ããåããã¨ã§ãã®ã§ã©ã¡ãã§ãè¯ãã§ãã</p>
<p>ã\(\,105°=\color{red}{45°}+\color{blue}{60°}\,\)<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama026.jpg" alt="" width="279" height="182" class="aligncenter size-full wp-image-13209" />\(\,45°\,\)ã®ä½å³ã§ãã<br />
\(\,\mathrm{AB}\,\)ã延é·ãã¦ã\(\,\mathrm{A}\,\)ãéãåç·ãå¼ãããã®\(\,90°\,\)ã®è§ã®äºçåç·ãæãã¾ããã</p>
<p>å¾ã¯\(\,45°\,\)ã®ç·ã®ä¸é¨ã\(\,1\,\)辺ã¨ããæ£ä¸è§å½¢ãæãã°\(\,60°\,\)ãå ããã¾ãã<br />
\(\,1\,\)辺ã¯é©å½ã«æ±ºãã¦è¯ãã§ããã<span class="red b">åå¾ãçãããã¨ãå©ç¨</span>ããã¨ã¯ããã§ããã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama027.jpg" alt="" width="278" height="212" class="aligncenter size-full wp-image-13210" />æ£ä¸è§å½¢ã¯åå¾\(\,2\,\)ã¤ã¨å¼¦ã§ã§ãã¦ãã¾ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama0278.jpg" alt="" width="257" height="203" class="aligncenter size-full wp-image-13211" /><br />
ã\(\,105°=\color{blue}{60°}ï¼\color{red}{45°}\,\)</p>
<p>\(\,60°\,\)ãä½ãããã«æ£ä¸è§å½¢ãå ã«æãã¨ã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama028.jpg" alt="" width="210" height="186" class="aligncenter size-full wp-image-13212" />\(\,1\,\)辺ã\(\,\mathrm{AB}\,\)ã®æ£ä¸è§å½¢ãæããã¨ã§\(\,60°\,\)ãä½å³ãã¾ããã</p>
<p>次ã«ããã延é·ãã¦\(\,\mathrm{A}\,\)ããåç·ãæããè§ã®äºçåç·ãæ¸ã足ãã¾ãã<br />
<img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama030.jpg" alt="" width="285" height="239" class="aligncenter size-full wp-image-13214" />\(\,\mathrm{A}\,\)ããåç·ãå¼ããã¨ãã®å¼§ãå©ç¨ãã¦è§ã®äºçåç·ãæãã¾ããã</p>
<h4>å¹³è¡å辺形ã«ãªãæ¡ä»¶ã¯åè£ï¼ã¤</h4>
<p>(4)<br />
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æç§æ¸ã«ããã®ã§ç¢ºèªãã¦ããã¦ãã ããã</p>
<p>ãã®5ã¤ã®æ¡ä»¶ãããã«ããããã©ããã¯ã<br />
æ°å¦ã®è¨èã®éè¦æ§ãç解ãã¦ãããã©ããã測ãã®ã«é©ãã¦ãã¾ãã</p>
<p>ããã«è¨ããã°è¯ãã§ãããè¨ããªã人ã¯æ°å¦ãè¦æã«æãã¦ããã®ã§ã¯ãªãã§ããããã<br />
æ°å¦ã®å¤§åãªãã¤ã³ããåãã£ã¦ããªãããã§ããã</p>
<p>⇒ã<a href="https://fromhimuka.com/oboetaro.html" rel="noopener" target="_blank">ä¸å¦æ°å¦ã®çææ»ç¥ææãè¦ã太éã</a></p>
<p>é«æ ¡ã®æ°å¦ãå«ãã§ã¾ããã©ãä¸å¦æ°å¦ã®ãã¹ã¦ã¯ããã«ããã¾ããï¼å®£ä¼ã§ããç¬ï¼<br />
ãã®éå»åããã¹ã¦ãè¦ã太éãã®ä¸ãã解説ãã¦ãã¾ãã</p>
<p>ãã¦ã宣ä¼ãçµãã£ãã¨ããã§è¨¼æãã¾ãããã<br />
証æã¯ç°¡åã§ãã</p>
<p>å³å½¢ã®è¨¼æã¯å³ã®ä¸ã§å®æããã¦ãããã¨ããã¤ã³ãã§ãã<br />
å³ã®ä¸ã§ç¤ºãã¦ããªãã®ã«ãæ¥æ¬èªã®æç« ã«ã§ããããããªãã§ãããã</p>
<p>ã\(\color{red}{\fbox{æ¡ä»¶}}\)</p>
<p>ã\(\,\mathrm{ABCD}\,\)ã¯å¹³è¡å辺形<br />
ã対è§ç·ã®äº¤ç¹ã¯\(\,\mathrm{O}\,\)<br />
ã\(\,\mathrm{E,F}\,\)ã¯ç·å\(\,\mathrm{OA,OC}\,\)ä¸ã®ç¹ã§\(\,\mathrm{AE=CF}\,\)</p>
<p><img src="https://arakannotie.com/wp-content/uploads/2019/03/2019saitama031.jpg" alt="" width="334" height="190" class="aligncenter size-full wp-image-13216" />証æã¯çµãã£ã¦ã¾ããã</p>
<p>ï¼è¨¼æï¼<br />
ã\(\,\mathrm{ABCD}\,\)ã¯å¹³è¡å辺形ã§å¯¾è§ç·ã¯ããããã®ä¸ç¹ã§äº¤ããã®ã§<br />
ãã\(\mathrm{\color{red}{AO}=\color{red}{CO}}ã･･･â \)<br />
ãã\(\mathrm{\color{blue}{BO}=\color{blue}{DO}}ã･･･â¡\)<br />
ã¾ã<br />
ãã\(\mathrm{EO=\color{red}{AO}-\color{magenta}{AE}}\)<br />
ãã\(\mathrm{FO=\color{red}{CO}-\color{magenta}{CF}}\)<br />
ä»®å®ãã\(\,\mathrm{\color{magenta}{AE}=\color{magenta}{CF}}\,\)ã§ããããâ ã¨åããã¨<br />
ãã\(\,\mathrm{EO=FO}ã･･･â¢\,\)<br />
â¡â¢ãã対è§ç·ãããããã®ä¸ç¹ã§äº¤ãã£ã¦ãããã¨ã示ãã¦ããã®ã§<br />
ãåè§å½¢\(\,\mathrm{EBFD}\,\)ã¯å¹³è¡å辺形ã§ãããï¼çµããï¼</p>
<p>å¹³è¡å辺形ã®æç«æ¡ä»¶ã§\(\,7\,\)ç¹ã£ã¦ããããããã§ããã</p>
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